The Hessian Form of an Elliptic Curve

  • N. P. Smart
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2162)

Abstract

In this paper we use the Hessian form of an elliptic curve and show that it offers some performance advantages over the standard representation. In particular when a processor allows the evaluation of a number of field multiplications in parallel (either via separate ALU’s, a SIMD type operation or a pipelined multiplication unit) one can obtain a performance advantage of around forty percent.

References

  1. 1.
    J.W.S. Cassels. Lectures on Elliptic Curves. LMS Student Texts, Cambridge University Press, 1991.Google Scholar
  2. 2.
    D.V. Chudnovsky and G.V. Chudnovsky. Sequences of numbers generated by addition in formal groups and new primality and factorisation tests. Adv. in Appl. Math., 7, 385–434, 1987.MathSciNetCrossRefGoogle Scholar
  3. 3.
    C. Clapp. Instruction level parallelism in AES Candidates. Second Advanced Encryption Standard Candidate Conference, Rome March 1999.Google Scholar
  4. 4.
    H. Cohen, A. Miyaji and T. Ono. Efficient elliptic curve exponentiation using mixed coordinates. In Advances in Cryptology, ASIACRYPT 98. Springer-Verlag, LNCS 1514, 51–65, 1998.CrossRefGoogle Scholar
  5. 5.
    M. Desboves. Résolution en nombres entiers et sous sa forme la plus générale, de l’équation cubique, homogéne, á trois inconnues. Nouvelles Ann. de Math., 45, 545–579, 1886.Google Scholar
  6. 6.
    C.K. Koc, T. Acer and B.S. Kaliski Jnr. Analyzing and comparing Montgomery multiplication algorithm. IEEE Micro, 16, 26–33, June 1996.Google Scholar
  7. 7.
    J. López and R. Dahab. Improved algorithms for elliptic curve arithmetic in GF(2n) In Selected Areas in Cryptography-SAC’ 98, Springer-Verlag, LNCS 1556, 201–212, 1999.CrossRefGoogle Scholar
  8. 8.
    G. Orlando and C. Paar. A high-performance reconfigurable elliptic curve processor for GF(2m). In Cryptographic Hardware and Embedded Systems (CHES) 2000, Springer-Verlag, LNCS 1965, 41–56, 2000.CrossRefGoogle Scholar
  9. 9.
    A.D. Woodbury, D.V. Bailey and C. Paar. Elliptic curve cryptography on smart cards without coprocessors. In Smart Card and Advanced Applications, CARDIS 2000, 71–92, Kluwer, 2000.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • N. P. Smart
    • 1
  1. 1.Dept. Computer ScienceUniversity of BristolBristol

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